The proof is similar to [1]}.
Since \(\phi \)  is even,
\(\phi (\xi ) = \sup \limits _{\eta \in \mathbb {R}^2,\,\phi ^o(\eta )=1} |\xi \cdot \eta |,\quad \xi \in \mathbb {R}^2,\) 
where \(\phi ^o\)  is the dual norm of \(\phi .\)  By the compactness of \(B^{\phi ^o}:=\lbrace \eta \in \mathbb {R}^2:\,\phi ^o(\eta )=1\rbrace ,\)  for any countable set \(\lbrace \eta _i\rbrace \)  dense in \(B^{\phi ^o}\)  and for any \(\mathcal {H}^1\) -rectifiable set \(K\subset \mathbb {R}^2\)
\(\phi (\nu _K(x)) = \sup \limits _{i\ge 1} |\nu _K(x)\cdot \eta _i|\quad \text{for $\mathcal {H}^1$-a.e. $x\in K.$}\) 
Hence, by [2]} for any open set \(U\subset \mathbb {R}^2\)
\(\int _{U\cap K} \phi (\nu _K)d\mathcal {H}^1 = \sup \limits _k \sup \Big \lbrace \sum \limits _{i=1}^k\int _{A_i\cap K} |\nu _K\cdot \eta _i|d\mathcal {H}^1:\,\text{$A_i\subset \subset U$ open and pairwise disjoint}\Big \rbrace .\) 
Moreover, by the area formula for any Borel set \(B\)
\(\int _{B\cap K} |\nu _K\cdot \xi |d\mathcal {H}^1 = |\xi | \int _{\pi _\xi (B)} \mathcal {H}^0(K\cap B_y^\xi )d\mathcal {H}^1(y),\) 
where
\(\pi _\xi (z) = z - \big (z\cdot \tfrac{\xi }{|\xi |}\big )\,\tfrac{\xi }{|\xi |}\) 
and given \(y\in \pi _\xi (B),\)  \(B_y^\xi :=\pi _\xi ^{-1}(y)\cap B.\)
Before the proof of the lemma let us recall some notation and results from [3]}. Given \(\xi \in \mathbb {R}^2\setminus \lbrace 0\rbrace ,\)  let \(\Pi _\xi :=\lbrace y\in \mathbb {R}^2:\,y\cdot \xi =0\rbrace .\)  For every set \(B\subset \mathbb {R}^2\)  and for every \(y\in \Pi _\xi \)  we define
\(B_y^\xi :=\lbrace t\in \mathbb {R}:\, y+t\xi \in B\rbrace .\) 
Moreover, for every \(u:B\rightarrow \mathbb {R}^2\)  we define \(\widehat{u}_y^\xi :B_y^\xi \rightarrow \mathbb {R}\)  by
\(\widehat{u}_y^\xi (t):= u(y + t\xi )\cdot \xi .\) 
When \(u\in GSBD^2(Q_1),\)  then \(\widehat{u}_y^\xi \in SBV_\mathrm {loc}^2([Q_1]_y^\xi )\)  for \(\mathcal {H}^1\) -a.e. \(\pi _\xi (Q_1)\)  and for all \(\xi \in \mathbb {R}^2\setminus \lbrace 0\rbrace .\)  In this case we define
\(I_y^\xi (u):=\int _{[Q_1]_y^\xi } |(\dot{u})_y^\xi |^2 dt,\) 
where \((\dot{u})_y^\xi \)  is the density of the absolutely continuity part of \(D \widehat{u}_y^\xi \)  and also
\(II_y^\xi (u):=|D(\tau (u\cdot \xi )_y^\xi )|([Q]_1^\xi ),\) 
where \(\tau (t):=\arctan (t).\)  Recall that
\(\int _{\Pi _\xi } I_y^\xi (u) \mathcal {H}^1(y) +\int _{\Pi _\xi } II_y^\xi (u) \mathcal {H}^1(y)\le \int _{Q_1} |e(u)|dx + \int _{Q_1} |e(u)|^2dx + \mathcal {H}^1(J_u)\) 
(see e.g. [3]}).
Step 1: There exists an at most countable set \(\subset B^{\phi ^o}\)  such that
\(\lim \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (J_{w_k})) =0\) 
for any \(\xi \in B^{\phi ^o}\setminus .\)
Indeed, let \(\)  be the set of all \(\xi \in B^{\phi ^o}\)  for which there exists \(y\in \pi _\xi (I_1)\)  such that \(\mathcal {H}^1(\pi _\xi ^{-1}(y)\cap \partial D_k)>0.\) 
By assumption (b) and Lemma REF  the set \(\)  is at most countable.
Let \(\lbrace w_{k_l}\rbrace \)  be arbitrary not relabelled subsequence of \(\lbrace w_k\rbrace .\)  In view of [3]} (applied with \(A=U_1^\infty \) ) for any \(\xi \in B^{\phi ^o}\setminus ,\)  \(\epsilon >0\)  and for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (Q_1)\)  there exists a further subsequence \(w_{k_{l_h}}\)  (possibly depending on \(\xi ,\)  \(\epsilon \)  and \(y\) )
\(\mathcal {H}^0(J_{[\widehat{w}_0]_y^\xi } \cap [Q_1\setminus U_1^\infty ]_y^\xi ) + \mathcal {H}^0([\partial U_1^\infty ]_y^\xi ) \le \liminf \limits _{h\rightarrow +\infty } \Big [\mathcal {H}^1(J_{[w_{k_{l_h}}]_y^\xi }) + \epsilon (I_y^\xi (w_{k_{l_h}}) + II_y^\xi {w_{k_{l_h}}})\Big ].\) 
By the definition of \(w_0\)  and \(U_1^\infty ,\)  the left-side of (REF ) is equal to 1 for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (I_1).\)  Theorefore, for such \(y\)  and for sufficiently small \(\epsilon >0\)  we have
\(\liminf \limits _{h\rightarrow +\infty } \mathcal {H}^1(J_{[w_{k_{l_h}}]_y^\xi })\ge 1.\)  Hence, for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (I_1)\)  the line \(\pi _\xi ^{-1}(y)\)  intersects \(J_{w_{k_{l_h}}}\)  for all \(h\)  and (REF ) follows.
Note that by [1]}
\(\liminf \limits _{k\rightarrow +\infty } \int _{Q_1\cap J_{w_k}} \phi (\nu _{J_{w_k}}) d\mathcal {H}^1 \ge \int _{I_1} \phi ({\bf e_2}) d\mathcal {H}^1.\) 
Step 2: Now we improve (REF ) by including coefficient 2 on the right-hand side of the inequality in the presence of a small error term.
We proceed in three substeps. We redefine the displacement \(w_k\)  in the convex envelope \(V_k^i\)  of each connected component \(K_k^i\)  of \(\partial D_k\)  in such a way that \(\partial V_k^i\)  become jump sets with the left-hand side of (REF ) lowered up to a small error.
Substep 2.1: First we identify \(\lbrace V_k^i\rbrace .\)
Fix any \(\delta \in (0,1).\) 
By (b) there exists \(k_\delta ^1>0\)  such that \(([-2,2]\times [-2,-\delta ]) \cup ([-2,2]\times [\delta ,2])\subset \mathrm {Int}(D_k)\)  for any \(k\ge k_\delta ^1.\)  Let \(F_k:=Q_1\cap D_k.\)  Note that \(\partial F_k \subset (Q_1\cap \partial D_k) \cup (\lbrace \pm 1\rbrace \times [-\delta ,\delta ])\)  and since \(D_k\in \mathcal {A}_{m_k},\)  the number of connected components \(\lbrace L_k^j\rbrace _{j\ge 1}\)  of \(\partial F_k\)  does not exceed \(m_k\) . Note that \(F_k\subset [-1,1]\times [-\delta ,\delta ]\)  and
\(\alpha _k: = & \int _{Q_1\cap \partial ^*E_{k}} \phi (\nu _{E_{k}}) d\mathcal {H}^1 + 2\int _{Q_1\cap E_{k}^{(1)} \cap \partial E_{k}} \phi (\nu _{E_{k}})d\mathcal {H}^1 \nonumber \\\ge & \int _{Q_2\cap \partial ^*F_k} \phi (\nu _{F_k}) d\mathcal {H}^1 + 2\int _{Q_2\cap F_k^{(1)} \cap \partial F_k} \phi (\nu _{F_k})d\mathcal {H}^1 - 4\delta \nonumber \\= & \sum \limits _{j\ge 1} \Big [\int _{Q_2\cap \partial ^*F_k\cap L_k^j} \phi (\nu _{F_k}) d\mathcal {H}^1 + 2\int _{Q_2\cap F_k^{(1)} \cap \partial F_k\cap L_k^j} \phi (\nu _{F_k})d\mathcal {H}^1\Big ] - 4\delta :=\alpha _k^{\prime }.\) 

Next repeating the same arguments of Step 1 in the proof of [1]} we can find a family \(\lbrace V_k^i\rbrace _i\)  of at most countably many pairwise disjoint closed convex sets with non-empty interior such that for each \(L_k^j\)  there exists a unique \(V_i\)  with \(L_k^j\subset V_k^i\)  and
\(\alpha _k^{\prime } \ge \sum \limits _{i\ge 1} \int _{\partial V_k^i} \phi (\nu _{V_k^i}^{}) d\mathcal {H}^1 - 6\delta \) 
see e.g. [1]}
Substep 2.2: Now we replace \(w_k\)  with another function \(v_k\)  associated to \(V_k^i.\)  Fix \(\xi _0\in \mathbb {R}^2\)  such that the jump set of the function
\(v_k:= w_k\chi _{Q_1\setminus \cup _i V_k^i} + \xi _0\chi _{\cup _i V_k^i}\) 
coincide with \(\cup _i \partial V_k^i\)  (up to a \(\mathcal {H}^1\) -negligible set).
By assumption (b) \(\cup _i\partial V_k^i \overset{K}{\rightarrow }I_1\)  as \(k\rightarrow +\infty .\)  Moreover, as in Step 1 we can find a countable set \(^{\prime }\subset B^{\phi ^o}\)  such that
by assumption (b) and (REF )
\(\limsup \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (\cup _i\partial V_k^i)) \le & \limsup \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (\partial D_k)) \nonumber \\\le &\lim \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (J_{w_k})) =0\) 

for all \(\xi \in B^{\phi ^o}\setminus (\cup ^{\prime }).\) 
Moreover, by assumption (d) \(v_k \rightarrow w_0\)  a.e. in \(Q_1\setminus U_1^\infty \)  and \(|v_k|\rightarrow +\infty \)  a.e. in \(U_1^\infty .\)
Substep 2.3: By convexity of each \(V_k^i\)  we observe that
\(\liminf \limits _{k\rightarrow +\infty } \mathcal {H}^0(\pi _\xi ^{-1}(y) \cap J_{v_k}) \ge 2 = 2\mathcal {H}^0(J_{[\widehat{w}_0]_y^\xi } \cap [Q_1\setminus U_1^\infty ]_y^\xi ).\) 
for all \(\xi \in B^{\phi ^o}\setminus (\cup ^{\prime })\)  and \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi .\) 
Thus, by repeating the arguments of Step 1 in the proof of [1]} we get
\(\liminf \limits _{k\rightarrow +\infty } \int _{\cup _i \partial V_k^i} \phi (\nu _{\cup _iV_k^i}^{})d \mathcal {H}^1 = \liminf \limits _{k\rightarrow +\infty } \int _{J_{v_k}} \phi (\nu _{J_{v_k}}^{})d \mathcal {H}^1 \ge 2\int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1,\) 
which together with (REF ) and (REF ) implies the assertion of the lemma.
